i) G i v e n e q u a t i o n i s x 2 + y 2 + 2 x − y − z + 3 C o m p l e t i n g t h e s q u a r e s w e g e t x 2 + 2 x + 1 + y 2 − 2 ( 1 2 ) ( y ) + 1 4 − 1 4 − z − 1 4 + 2 = 0 ⇒ ( x + 1 ) 2 + ( y − 1 2 ) 2 + 7 4 − z = 0 ⇒ z = ( x + 1 ) 2 + ( y − 1 2 ) 2 + 7 4 T h e g r a p h o f t h e g i v e n e q u a t i o n i s Given \ equation\ is \ x^2+y^2+2x-y-z+3\\
Completing \ the \ squares \ we \ get \
x^2+2x+1+y^2-2(\frac{1}{2})(y)+\frac{1}{4}-\frac{1}{4}-z-\frac{1}{4}+2=0\\
\Rightarrow\ (x+1)^2+(y-\frac{1}{2})^2+\frac{7}{4}-z=0\\
\Rightarrow \ z=(x+1)^2+(y-\frac{1}{2})^2+\frac{7}{4}\\
The\ graph\ of \ the \ given\ equation\ is G i v e n e q u a t i o n i s x 2 + y 2 + 2 x − y − z + 3 C o m pl e t in g t h e s q u a res w e g e t x 2 + 2 x + 1 + y 2 − 2 ( 2 1 ) ( y ) + 4 1 − 4 1 − z − 4 1 + 2 = 0 ⇒ ( x + 1 ) 2 + ( y − 2 1 ) 2 + 4 7 − z = 0 ⇒ z = ( x + 1 ) 2 + ( y − 2 1 ) 2 + 4 7 T h e g r a p h o f t h e g i v e n e q u a t i o n i s
ii) Given equation is
3 y 2 + 3 z 2 + 4 x + 3 y + z = 9 C o m p l e t i n g t h e s q u a r e s w e g e t 3 ( y + 1 2 ) 2 + 3 ( z + 1 6 ) 2 + 4 x = ( 59 6 ) 2 T h e g r a p h o f t h e g i v e n c u r v e i s 3y^2+3z^2+4x+3y+z=9\\
Completing\ the\ squares \ we\ get\\
3(y+\frac{1}{2})^2+3(z+\frac{1}{6})^2+4x=(\sqrt \frac{59}{6})^2\\
The\ graph \ of \ the\ given\ curve\ is \\ 3 y 2 + 3 z 2 + 4 x + 3 y + z = 9 C o m pl e t in g t h e s q u a res w e g e t 3 ( y + 2 1 ) 2 + 3 ( z + 6 1 ) 2 + 4 x = ( 6 59 ) 2 T h e g r a p h o f t h e g i v e n c u r v e i s
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